Basic Fractions (OCR GCSE Maths)

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Basic Fractions

What is a fraction?

  • A fraction is part of a whole
  • A fraction is written as a over b where a and b are whole numbers (integers)
    • The number on the top, a, is called the numerator
    • The number on the bottom, b, is called the denominator
  • a over b means split something into b parts and take a of them
    • 1 half means split something into 2 parts and take 1 of these parts
    • 2 over 3 means split the something into 3 parts and take 2 of these parts
    • 4 over 5 means split something into 5 parts and take 4 of these parts

 

How do I find equivalent fractions?

  • “Splitting something into 2 parts and taking 1 of these parts” gives the same overall amount as “splitting something into 4 parts and taking 2 of these parts”
    • 1 half is the same as 2 over 4 
    • Using the same idea, 2 over 4 is the same as 4 over 8, or  8 over 16, or 3 over 6 etc
  • Equivalent fractions are two fractions that represent the same amount (they are different ways of writing the same thing)
  • To make equivalent fractions, multiply the top and bottom of a fraction by the same amount
    • 5 over 6 is equivalent to fraction numerator 5 cross times 2 over denominator 6 cross times 2 end fraction equals 10 over 12 and fraction numerator 5 cross times 3 over denominator 6 cross times 3 end fraction equals 15 over 18 and fraction numerator 5 cross times 4 over denominator 6 cross times 4 end fraction equals 20 over 24 …etc
    • For each fraction, there are an infinite number of equivalent fractions
  • To cancel a fraction down to its “simplest” equivalent form, divide the top and bottom of a fraction by the biggest whole number that goes into both (“common factor”)
    • 12 over 18 equals fraction numerator 12 divided by 6 over denominator 18 divided by 6 end fraction equals 2 over 3
    • 25 over 45 equals fraction numerator 25 divided by 5 over denominator 45 divided by 5 end fraction equals 5 over 9

 

How do I find a fraction of an amount?

  • Method 1: divide by the denominator and multiply by the numerator
    • To find 1 half of an amount, divide it by 2
    • To find 1 third of an amount, divide it by 3
    • To find 2 over 5 of an amount, divide it by 5 then multiply it by 2
      • 2 over 5 of 60: do 60 ÷ 5 = 12, then 12 × 2 = 24
  • Method 2 (if you know how to change fractions into decimals): change the fraction into a decimal, then multiply
    • To find 1 fourth of an amount, multiply the amount by 0.25
    • To find 9 over 10 of an amount, multiply the amount by 0.9
  • Method 3 (if you know how to multiply fractions): write both numbers as fractions and multiply two fractions together
    • To find 2 over 3 of 60, work out 2 over 3 cross times 60 over 1
    • To find 5 over 6 of 8 over 3, work out 5 over 6 cross times 8 over 3

Mixed Numbers & Top Heavy Fractions

What are mixed numbers & top heavy fractions?

  • A mixed number has a whole number (integer) part and a fraction part

    eg. 3 3 over 4 has the whole number 3 and the fraction 3 over 4, meaning “three and three quarters”

  • A top heavy fraction – also called an improper fraction – is one with the top (numerator) bigger than the bottom (denominator)

    eg. 15 over 4 means “fifteen quarters”

Turning mixed numbers into top heavy fractions

  1. Multiply the whole number by the denominator (big × bottom)
  2. Add that value to the numerator 
  3. Write the "new" numerator over the same denominator as before 

Turning top heavy fractions into mixed numbers

  • Divide the top by the bottom (to get a whole number and a remainder)
  • The whole number is the big number
  • The fraction part is the remainder over the denominator

Exam Tip

  • Top heavy fractions are also called "improper" fractions

Worked example

(a) Write 5 3 over 4 as an improper fraction

Multiply the whole number by the denominator, and add to the numerator. Keep the denominator the same.

fraction numerator open parentheses 5 cross times 4 space close parentheses plus 3 over denominator 4 end fraction

Simplify

bold 23 over bold 4

(b) Write 17 over 5 as a mixed number

Divide the top by the bottom

17 ÷ 5 = 3 remainder 2

The final answer is 3, with 2 parts still left over to be divided by 5, which can be written as a mixed number

bold 3 bold 2 over bold 5

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Mark

Author: Mark

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.